The inexorable resistance of inertia determinesthe initial regime of drop coalescenceJoseph D. Paulsena,1, Justin C. Burtona, Sidney R. Nagela, Santosh Appathuraib,2,Michael T. Harrisb, and Osman A. Basaranb

aJames Franck Institute and Department of Physics, University of Chicago, Chicago, IL 60637; and bSchool of Chemical Engineering, Purdue University,West Lafayette, IN 47907

Edited by William R. Schowalter, Princeton University, Princeton, NJ, and approved February 28, 2012 (received for review December 19, 2011)

Drop coalescence is central to diverse processes involving disper-sions of drops in industrial, engineering, and scientific realms.During coalescence, two drops first touch and then merge as theliquid neck connecting them grows from initially microscopic scalesto a size comparable to the drop diameters. The curvature of theinterface is infinite at the point where the drops first make contact,and the flows that ensue as the two drops coalesce are intimatelycoupled to this singularity in the dynamics. Conventionally, thisprocess has been thought to have just two dynamical regimes: aviscous and an inertial regime with a cross-over region betweenthem. We use experiments and simulations to reveal that a thirdregime, one that describes the initial dynamics of coalescencefor all drop viscosities, has been missed. An argument based onforce balance allows the construction of a new coalescence phasediagram.

emulsions ∣ fluid singularity

The collision and coalescence of water drops, so essential toraindrop growth and the development of thunderstorms, have

captivated the attention of the atmospheric science communitysince the early studies by Benjamin Franklin and Lord Rayleigh(1). Coalescence also plays a central role in industrial processesinvolving emulsions or dispersions (2, 3). For example, in thepetroleum industry, coalescence occurs during dispersed waterremoval and during oil desalting (4). It is a dominant processin determining the shelf life of emulsion-based products suchas salad dressing and mayonnaise (5), and it occurs in dense spraysystems and combustion (6). Also, sintering of two spherical par-ticles closely resembles the coalescence of two dispersion drops inan emulsion (7). Moreover, the controlled coalescence of dropsin microfluidic devices promises a host of potential applicationsin chemistry, biochemistry, and materials science (8).

The initial dynamics of coalescence are expected to be univer-sal. The expansion of the liquid neck connecting two drops iscontrolled by the Laplace pressure, which diverges when the cur-vature of the liquid interface is infinite at the point where thedrops first touch. Thus the change in topology, as two drops be-come one, is inextricably linked to a singularity in the dynamics.Different regimes of coalescence have been studied (9–25). Theunderstanding that has emerged is that coalescence has just twodynamical regimes with a cross-over region between them: a vis-cous regime, which always dominates at sufficiently early timeswhen the neck radius is microscopically small, and an inertial re-gime that occurs at late times if viscous effects become negligible.We use experiments and simulations to show that a third regime,one that describes the true initial dynamics of coalescence, hasbeen missed. We present our results in terms of a phase diagramof coalescence that shows the three distinct dynamical regimes.

Results and DiscussionIn the experiment, two pendant drops with radii A ≈ 0.1 cm aresuspended as in Fig. 1A from nozzles and slowly translated untilthey touch at their equators. Except where otherwise stated, thedrops are silicone oil (surface tension γ ¼ 20 dyn∕cm and density

ρ ¼ 0.97 g∕cm3). The liquid viscosity, μ, or equivalently the di-mensionless Ohnesorge number, Oh ¼ μ∕

ffiffiffiffiffiffiffiffiffiργA

p, is varied. We

use a high-speed camera and electrical resistance measurements(24) to capture the dynamics.

In the simulations, two isolated spherical drops of radiusA areconnected by a small neck of radius rmin ¼ 0.001A. The dynamicsthat ensue are determined by solving the full Navier–Stokes equa-tions by a finite-element algorithm that we have previously usedsuccessfully to study diverse situations involving drop breakup(26, 27). Creeping-flow simulations, where inertia is neglected,are also performed.

Fig. 1A shows two drops at the instant of contact, τ ¼ 0. Todistinguish local versus global motion during merging, we sub-tract an image taken after the neck has grown to a small size fromthe one at τ ¼ 0 as shown in Fig. 1 B and C for fluid viscositiesμ ¼ 58;000 cP (Oh ¼ 440) and 49 cP (Oh ¼ 0.32), respectively.At high viscosity, the two drops move together rigidly whereas atlower viscosity, the only appreciable motion occurs near thewidening neck. This qualitative difference heralds the existenceof a different and distinct regime.

The transition between these regimes can be understood froma force-balance argument that is based on the observation that inthe perfectly viscous (i.e., Stokes) regime, the drops outside theimmediate vicinity of the neck are rigidly translated towards eachother. This was shown by Hopper (9, 10) in an exact analytic so-lution of coalescence in two dimensions (2D); this global motionwas also seen in 3D Stokes simulations (11) and the early-timeasymptotic behavior was later analytically extended to three di-mensions (3D) (12). To be in the Stokes regime, therefore, theforce of the neck pulling the two drops together must be suffi-ciently large to produce the required center-of-mass accelerationof each drop (i.e., the fluid on each side of the z ¼ 0 plane). Theasymptotic acceleration (9) is ac:o:m: ¼ γ2½lnðrmin∕8AÞ�2∕2π2μ2A.

The coalescence is driven by surface tension. Therefore, anupper bound for the inward force of the neck on the drops is givenby the surface-tension force, Fγ, around the circumference of theneck at its minimum radius. In 3D: Fγ ¼ 2πγrmin. If Fγ is toosmall to translate the drops, each having a mass m ¼ 4

3πA3ρ,

the flows cannot be in the Stokes regime. Therefore, the Stokesregime can only be achieved when Fγ ≳mac:o:m: leading to thethreshold criterion for entering the Stokes regime:

Oh ∝����ln

�1

8

rmin

A

������rmin

A

�−1∕2

: [1]

Author contributions: J.D.P., J.C.B., S.R.N., S.A., M.T.H., and O.A.B. designed research;J.D.P. and S.A. performed research; J.D.P., J.C.B., and S.R.N. conducted the experiments;S.A., M.T.H., and O.A.B. conducted the simulations; J.D.P., J.C.B., S.R.N., S.A., M.T.H., andO.A.B. analyzed data; and J.D.P., J.C.B., S.R.N., S.A., M.T.H., and O.A.B. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence regarding experiments should be addressed. E-mail: paulsenj@uchicago.edu.

2To whom correspondence regarding simulations should be addressed. E-mail: santosha@purdue.edu.

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Therefore, for 3D drops of finite viscosity, the asymptoticdynamics, in the limit rmin∕A → 0, can never be in the Stokesregime. Below this threshold, Fγ is balanced by the inertia ofthe drops,mac:o:m:, and the dynamics are governed by local flows.

An analogous argument in 2D (where Fγ ¼ 2γ independent ofrmin, and m ¼ πA2ρ) suggests a phase boundary for 2D drops:Oh ∝ j lnðrmin∕8AÞj∕ ffiffiffiffiffiffi

4πp

. Thus the logarithmic divergence ofac:o:m: also precludes the Stokes regime in 2D until rmin∕A growsto a sufficient size.

Fig. 2A presents a phase diagram of the coalescence regimesfor 3D drops, which includes the inertial and Stokes regimes inaddition to this “inertially limited viscous” regime. In this regime,inertia and viscosity play a role in the dynamics; the inertia asso-ciated with each drop moving as a rigid object precludes thesystem from being in the Stokes regime. The inertially limitedviscous to inertial cross-over was previously determined (24) to bermin∕A ∝ Oh, in contrast to earlier work that had suggested thatthis cross-over occurs at rmin∕A ∝ Oh2 (12, 17, 18). (Previously,there were believed to be only two coalescence regimes—a vis-cous one and an inertial one—so this cross-over is referred toas the viscous-to-inertial cross-over in the literature.) The iner-tially limited viscous to Stokes transition is described by 1. Weemphasize that, contrary to earlier studies, we find the Stokesand inertial regimes do not share a phase boundary; they are bothpreceded by the inertially limited viscous regime. Thus, at earlytimes, a model of pure Stokes flow for the coalescence of spheresis never valid. We note that this is reminiscent of the singularity indrop breakup (28), where there are also three regimes, and theStokes regime does not extend to rmin → 0.

Having argued for the distinct identities of the inertially lim-ited viscous regime versus the Stokes regime on theoreticalgrounds, we now offer evidence from experiment and simulationthat these regimes are, in fact, different. First, we probe the globalmotion of the drops by measuring the velocity of the back of onedrop, vb:o:d:. Since this point is the farthest from the singularity, itisolates the global motion from the flow near the growing neck.

For 3D drops in the inertially limited viscous regime, forcebalance gives ac:o:m: ≈ Fγ∕m ¼ 3γrmin∕2A3ρ. Using rmin ¼ τγ∕μas seen in Fig. 3E consistent with previous experiments (17, 18,20, 24, 25), we integrate to get

vb:o:d: ≈3γ2

4μA3ρτ2 ¼ 3μ

4A3ρr2min: [2]

If Oh > 1, then the flows eventually enter the Stokes regime,where to first order

vb:o:d: ≈γ

2πμArmin

����ln�1

8

rmin

A

�����: [3]

We find that the creeping-flow simulation follows the exactanalytic 2D Stokes solution (3). In Fig. 2B, we plot vb:o:d: for dropsof finite viscosity in the simulation and experiment. The curvesexhibit the predicted superlinear growth of vb:o:d: at early timesuntil the velocities merge onto the Stokes curve. The data showexceptional agreement between simulation and experiment.

CBA

µ = 58,000 cP (Oh = 440) µ = 49 cP (Oh = 0.32)τ = 0

0.5 mm

Fig. 1. Coalescence of silicone-oil drops with viscosities μ ¼ 58;000 cP (Oh ¼ 440) and 49 cP (Oh ¼ 0.32). (A) Two pendant drops at the instant they contact,τ ¼ 0. (The two bright spots are from back lighting.) We subtract an image taken after the neck has grown to a size rmin ¼ 0.25A from the one at τ ¼ 0 for B,μ ¼ 58;000 cP and C, 49 cP.

rmin /A

v b.o

.d.

Simulation:

Experiment:Oh=326.63.41.20.660.32

Creeping FlowOh=60402010

10–3 10–2 10–1 10010–3

10–2

10–1

100

101

102 Stokes regime

Inertial regime

Iviscous regime

rmin /A

Oh

=

A

B

Fig. 2. Phase diagram for 3D coalescence. (A) While the inertial regime (12–24) and the Stokes regime (9, 10, 11, 12, 16) have been established in recentyears, the inertially limited viscous regime is identified by this work. The in-ertially limited viscous to inertial cross-over was recently determined by ex-periments (open circles) and a scaling argument (solid line) (24). Here, weidentify the inertially limited viscous to Stokes cross-over with a force-balanceargument (dashed line: 1 with a proportionality constant of 1.4), simulations(filled triangles), and experiments (open triangles). The data depart from theprediction at large rmin, where we expect finite size effects to enter. (B) Toobserve the inertially limited viscous to Stokes cross-over, we measure vb:o:d:versus rmin∕A over a range of viscosities from simulation (solid lines) and ex-periment (symbols). The velocities fall onto the creeping-flow curve at largermin∕A but peel away at smaller rmin∕A. The velocity scaling at small rmin∕A isqualitatively captured by 2. The cross-over neck radius at which the macro-scopic drop velocity vb:o:d: merges onto the Stokes solution is plotted in A, forthe viscosities that exhibit such a cross-over within our range of data.

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We measure the cross-over neck radius at which the macro-scopic drop velocity vb:o:d: merges onto the Stokes solution.We plotOh versus rmin∕A as the threshold for entering the Stokesregime on the phase diagram for 3D coalescence (Fig. 2A). Athigher viscosities, the linear (Stokes) regime is entered at smallerrmin. The data agree well with the prediction from 1.

The neck shapes also differ between the Stokes and the inter-tially limited viscous regimes. Because the exact analytic 2D solu-tion extends over the entire domain of rmin (whereas the 3Dsolution only exists for small rmin), we use it here as a convenientway to account approximately for finite-size effects and to com-pare our experiment and simulation. In Fig. 3 A and B, we plotthis 2D Stokes solution (9) in the neck region against experimentand simulation, for Oh ≫ 1 and Oh ¼ 0.6, in the r-z plane (withthe origin at the initial point of contact). The Stokes solutionagrees with the high-viscosity data, but it clearly fails to fit theshapes at Oh ¼ 0.6 where both experiment and simulationshow a much broader neck. In particular, the 2D analytic Stokessolution has a maximum neck curvature, κ, that obeys 1∕κA ¼14ðrmin∕AÞ3 to first order, which we find is in good agreement with

our data in the Stokes regime (i.e.,Oh ≫ 1). In the inertially lim-ited viscous regime (i.e., Oh approximately 1), we also find that

1∕κA scales as ðrmin∕AÞ3 but with a significantly larger prefactor(approximately 1.2 instead of 1

4).

The dynamics also differ between these two regimes. For high-viscosity drops, measurements of the neck radius rmin versus timein the experiment and simulation are consistent with the exactanalytic 2D Stokes solution (Fig. 3D).* For lower-viscosity drops,the 2D Stokes solution does not fit the data (Fig. 3E). Instead, theneck radius grows linearly with time, consistent with rmin ¼ τγ∕μ,a form that one might guess from dimensional analysis alone.(For the experimental data in Fig. 3E, we coalesce hemisphericaldrops attached to circular nozzles separated by a distance 2A.This altered boundary condition does not affect our results: Usinghigh-speed imaging, we find that for rmin ≪ A, the dynamics areinsensitive to this change in boundary conditions in both the in-ertially limited viscous and Stokes regimes.) This linear growthhas been observed in previous experiments, but has incorrectlybeen assumed to be the dynamics of Stokes coalescence (17,18, 20, 24, 25) and therefore was not recognized as evidence

0.02 20.020.0- -0.02

0.02

0.08

0.05

0.02

0.08

0.05

0 0

10–4 10–3 10–2 10–1 10010–4

10–3

10–2

10–1

100

10–2 10–1 10010–2

10–1

100

G H

z/A z/A

r/A

z/A z/A

r/A

r min

/A

F

z/A

z/A

I

D E

A B C

0.02-0.02

0.02

0.08

0.05

0 40.040.040.0 000

Fig. 3. Coalescence dynamics in the Stokes regime (Left: creeping-flow simulation, Oh ¼ 440 experiment) versus the inertially limited viscous regime (Center:Oh ¼ 0.6 simulation and experiment) and the inertial regime (Right: Oh ¼ 0.007 simulation and experiment). (A,B,C) Neck profiles from simulation (solid lines)and experiment (symbols) at three different times, compared with the 2D Stokes theory in A and B [dashed lines (9)]. The Stokes profiles agree with high-viscosity data (A) but do not capture the broader interfacial shapes at Oh ¼ 0.6 (B). (D,E,F) rmin∕A versus rescaled time in the simulation (solid lines) andexperiment (circles). High-viscosity drops (D) follow with the Stokes theory (dashed line). At intermediate viscosity (E), rmin∕A does not agree with the Stokestheory but instead grows at the viscous-capillary velocity, rmin ¼ τγ∕μ (dotted line). At low viscosity in the inertial regime (F), rmin∕A ¼ 1.4τ1∕2ðγ∕ρA3Þ1∕4 (dash-dot line). Note that in E, we begin plotting the simulation data at τγ∕μA ¼ 6 × 10−3, where we estimate that transients from the initial conditions have decayed.The experimental data in E were obtained by a high-speed electrical method on glycerol-salt-water drops (μ ¼ 230 cP, γ ¼ 65 dyn∕cm, ρ ¼ 1.2 g∕cm3, andA ¼ 0.2 cm) (24). (G,H,I) Instantaneous streamlines from simulations at rmin ¼ 0.03A in the (G) Stokes regime (creeping-flow simulation), (H) the inertiallylimited viscous regime (Oh ¼ 0.6), and (I) the inertial regime (Oh ¼ 0.007). The flows are qualitatively different in all three regimes.

*Previous viscous coalescence experiments (16, 17, 18, 24, 25) have compared the neckradius rminðτÞ against the theoretical prediction (12), rmin ≈ τγj lnðτγ∕μAÞj∕πμ. This approx-imate form breaks down (12) for rmin > 0.03A. Therefore, we compare our measurementsagainst the full analytic 2D solution (9), which can be done over the entire range of data.

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of a separate regime. We emphasize that the observed power law,rmin ∝ τ, is different from the inertial scaling where rmin ∝ τ1∕2 [asshown in Fig. 3F and by previous work (12, 13, 14, 15)], whichdemonstrates that the inertially limited viscous regime is distinctfrom the inertial regime as well.

Lastly, our simulations give the flow profiles near the singular-ity in the Stokes and the inertially limited viscous regimes (Fig. 3G and H). The flow is expected to occur over a length scale com-parable to rmin in the Stokes regime (12) and over a length scaler2min∕A in the inertially limited viscous regime (24). Indeed,whereas features in the creeping-flow streamlines are roughly thesize of rmin, the streamlines at intermediate Oh exhibit recircula-tion zones, which constrict the flows near the neck. Comparingthe geometry of the streamlines further solidifies that the iner-tially limited viscous regime (Fig. 3 B,E, and H) is distinct fromthe one described by pure Stokes flow (Fig. 3 A,D, andG) and theone described by inertial flow (Fig. 3 C,F, and I). The streamlinesin Fig. 3I corroborate the definition of the Reynolds numberproposed in ref. 24 that dictates the inertially limited viscousto inertial cross-over.

ConclusionAs two drops begin to coalesce and a microscopic liquid neckforms between them, the curvature of the interface and theLaplace pressure that develops due to surface tension bothdiverge at the instant when the drops first touch. In drop coales-cence with no external fluid, previous work (12) incorrectly ledto the conclusion that only viscous forces, along with surfacetension, should dominate on small scales, a dynamical regimereferred to as the Stokes regime. Our work identifies a necessarycondition for Stokes flow to occur. The dynamics cannot be in theStokes regime until the surface-tension force around the neck islarge enough to rigidly translate the two initially stationary dropstowards each other. The inexorable resistance of inertia rears itshead at even these small scales.

Therefore, a heretofore unknown dynamical regime controlsthe singularity at early times for drops of any viscosity. In thisinitial, asymptotic regime of drop coalescence in air, all of theunderlying forces, that is inertial, viscous, and surface-tensionforces, are important. Hence, the two dynamical regimes referredto as the Stokes regime, where inertia is negligible, and the iner-tial regime, where viscous force is negligible, can only be attainedonce the neck has grown to a sufficient size. Once the drop hasentered the Stokes or inertial regimes, our measurements areconsistent with the earlier predictions for the dynamics in thoseregimes.

A dynamically similar response is observed when a liquid fila-ment breaks in air. At small neck size, the viscous and inertialregimes both give way to a third regime, where inertial, viscous,and surface-tension forces are all important (28). When a liquidfilament breaks in another liquid, however, the dynamics arequalitatively different: In that case the asymptotic dynamics ofthinning may occur in the absence of inertia (29, 30). Further in-sight may likewise be gained by studying drop coalescence inside

a second immiscible liquid, which is a problem of immense prac-tical importance (31, 32).

Materials and MethodsExperiment. High-speed imaging and electrical measurements were sepa-rately performed to capture the coalescence dynamics of isolated liquiddrops in air. In high-speed imaging measurements, two silicone-oil drops withradii A ≈ 0.1 cm were suspended side by side as in Fig. 1A from syringe nee-dles. Different silicone oils were used to vary the liquid viscosity, μ, from 49 to58,000 cP, while keeping other fluid parameters constant (surface tension,γ ¼ 20 dyn∕cm and density, ρ ¼ 0.96 to 0.98 g∕cm3). Because they are highlywetting, pendant silicone-oil drops tend to climb up stainless steel syringeneedles until the needles protrude from the bottoms of the drops. To preventthis, the needles were treated with an electronic coating (Novec EGC-1700,3M) that inhibits wetting. The drops were aligned and then slowly translatedwith a micrometer stage until they gently touched at their equators. The re-sulting coalescence dynamics were recorded with a high-speed digital camera(Phantom v12, Vision Research).

The electrical method is described in detail in ref. 24. The experimentaldata in Fig. 3E were obtained by this method on glycerol-salt-water drops(μ ¼ 230 cP, γ ¼ 65 dyn∕cm, ρ ¼ 1.2 g∕cm3, and A ¼ 0.2 cm).

Simulation. The coalescence of two identical, isolated spherical drops ofradius A of an incompressible Newtonian fluid that are surrounded by a dy-namically passive gas is simulated by connecting them with a small bridge ofradius rmin (typically equal to 0.1% of the drop radius) and height zmin ≪ rmin.The ensuing coalescence dynamics are governed by the continuity and theNavier–Stokes equations, i.e., the Navier–Stokes system. Because the twodrops are identical and the two-drop configuration is axially symmetric,the computational domain is the planar quadrant that consists of one ofthe drops and one-half of the bridge that is bounded by the plane of sym-metry, the axis of symmetry, and the liquid–gas (L-G) interface. The Navier–Stokes system is solved subject to symmetry boundary conditions along theplane of symmetry and the axis of symmetry, and the kinematic and tractionboundary conditions along the L-G interface (26, 27). This free boundaryproblem is solved numerically by a fully implicit method of lines arbitrary La-grangian–Eulerian algorithm that uses the Galerkin/finite-element method(G/FEM) for spatial discretization and an adaptive finite difference method(FDM) for time integration (26, 27). On account of the free boundary natureof the problem, the interior of the flow domain is discretized by an adaptiveelliptic mesh generation algorithm (33).

The G/FEM converts the transient system of nonlinear partial differentialequations to a system of nonlinear ordinary differential equations (ODEs).The FDM time integrator reduces the system of ODEs to a large system ofnonlinear algebraic equations. This system of equations is then solved byNewton’s method with an analytically calculated Jacobian.

Starting from an initially quiescent state, the dynamics are followed untilthe two drops have coalesced into one and the dynamics have ceased. Simu-lations are carried out for both situations in which inertia is present, i.e.,Oh isfinite, and also when inertia is negligible, i.e, 1∕Oh ¼ 0, such that the dropsundergo creeping (Stokes) flow.

ACKNOWLEDGMENTS. We thank Michelle Driscoll, Efi Efrati, and WendyZhang. This work was supported by National Science Foundation (NSF) GrantDMR-1105145, the University of Chicago NSF Materials Research Science andEngineering Centers DMR-0820054, the NSF Engineering Research Center forStructured Organic Particulate Systems (EEC-0540855), and the Basic EnergySciences Program of the US Department of Energy. Use of facilities of theKeck Initiative for Ultrafast Imaging is gratefully acknowledged.

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